[PPT] - Selfish Routing and the Price of Anarchy Tim Roughgarden Cornell PowerPoint Presentation

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Selfish Routing and the Price of Anarchy

Tim Roughgarden Cornell University

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Algorithms for Self-Interested Agents

Our focus: problems in which multiple

agents (people, computers, etc.) interact

Motivation: the Internet

decentralized operation and ownership

Traditional algorithmic approach:

agents classified as obedient or adversarial

– examples: distributed algorithms, cryptography

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Algorithms and Game Theory

Recent trend: agents have own independent

bjectives (and act accordingly)

New goal: algorithms that account for

strategic behavior by self-interested agents

Natural tool: game theory

theory of “rational behavior” in competitive,

collaborative settings

– [von Neumann/Morgenstern 44]

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Objectives

This talk: understand consequences of noncooperative behavior

when is the cost of selfish behavior severe?

– the “price of anarchy” [Koutsoupias/Papadimitriou 99]

what can we do about it?

– design strategies, economic incentives

Our setting: routing in a congested network

will focus on [Roughgarden/Tardos FOCS ’00/JACM ’02]
and also [Roughgarden STOC ’02/JCSS to appear]

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Motivating Example

Example: one unit of traffic wants to go from

s to t

Question: what will selfish network users do?

assume everyone wants smallest-possible delay

s t

ℓ(x)=x ℓ(x)=1 delay depends on congestion no congestion effects

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Motivating Example

Claim: all traffic will take the top link. Reason:

Є > 0 ⇒ traffic on bottom is envious
Є = 0 ⇒ envy-free outcome

– all traffic incurs one unit of delay

s t

ℓ(x)=x ℓ(x)=1 Flow = 1-Є Flow = Є this flow is envious!

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Can We Do Better?

Consider instead: traffic split equally Improvement:

half of traffic has delay 1 (same as before)
half of traffic has delay ½ (much improved!)

s t

ℓ(x)=x ℓ(x)=1 Flow = ½ Flow = ½

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Braess’s Paradox

Initial Network:

s t x 1 ½ x 1 ½ ½ ½

Delay = 1.5

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Braess’s Paradox

Initial Network: Augmented Network:

s t x 1 ½ x 1 ½ ½ ½

Delay = 1.5

s t x 1 ½ x 1 ½ ½ ½

Now what?

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Braess’s Paradox

Initial Network: Augmented Network:

s t x 1 ½ x 1 ½ ½ ½

Delay = 1.5 Delay = 2

s t x 1 x 1

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Braess’s Paradox

Initial Network: Augmented Network: All traffic incurs more delay! [Braess 68]

also has physical analogs [Cohen/Horowitz 91]

s t x 1 ½ x 1 ½ ½ ½

Delay = 1.5 Delay = 2

s t x 1 x 1

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The Mathematical Model

a directed graph G = (V,E)
k source-destination pairs (s1 ,t1), …, (sk ,tk)
a rate ri of traffic from si to ti
for each edge e, a latency function ℓe(•)

– assumed continuous and nondecreasing s1 t1 ℓ(x)=x

Flow = ½ Flow = ½

ℓ(x)=1 Example: (k,r=1)

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Routings of Traffic

Traffic and Flows:

fP = amount of traffic routed on si-ti path P
flow vector f

routing of traffic Selfish routing: what flows arise as the routes chosen by many noncooperative agents?

s t

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Nash Flows

Some assumptions:

agents small relative to network
want to minimize personal latency

Def: A flow is at Nash equilibrium (or is a Nash flow) if all flow is routed on min-latency paths [given current edge congestion]

x

s t

1

Flow = .5 Flow = .5

s t

1

Flow = 0 Flow = 1

x

Example:

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Some History

traffic model, definition of Nash flows

given by [Wardrop 52]

– historically called user-optimal/user equilibrium

Nash flows exist, are (essentially) unique

– due to [Beckmann et al. 56]

Nash flows also arise via distributed

shortest-path protocols (e.g., OSPF, BGP)

– as long as latency used for edge weights – convergence studied in [Tsitsiklis/Bertsekas 86]

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The Cost of a Flow

Def: the cost C(f) of flow f = sum of all delays incurred by traffic (aka total latency)

s t x 1

½ ½ Cost = ½•½ +½•1 = ¾

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The Cost of a Flow

Def: the cost C(f) of flow f = sum of all delays incurred by traffic (aka total latency) Formally: if ℓP(f) = sum of latencies of edges

f P (w.r.t. the flow f), then:

C(f) = ΣP fP • ℓP(f)

s t s t x 1

½ ½ Cost = ½•½ +½•1 = ¾

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Inefficiency of Nash Flows

Note: Nash flows do not minimize total latency

observed informally by [Pigou 1920]
lack of coordination leads to inefficiency
Cost of Nash flow = 1•1 + 0•1 = 1
Cost of optimal (min-cost) flow = ½•½ +½•1 = ¾

s t x 1

1

½ ½

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How Bad Is Selfish Routing?

s t x 1

1

½ ½

Pigou’s example is simple…

Central question: How inefficient are Nash

flows in more realistic networks?

Goal: prove that Nash flows are near-optimal

want laissez-faire approach to managing networks

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The Bad News

Bad Example:

(r = 1, d large) Nash flow has cost 1, min cost ≈ 0

⇒ Nash flow can cost arbitrarily more than

the optimal (min-cost) flow

– even if latency functions are polynomials s t xd 1

1 1-Є

Є

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Hardware Offsets Selfishness

Approach #1: use different type of guarantee

Theorem: [Roughgarden/Tardos 00] for every

network: ≤ Nash cost at rate r

pt cost at rate 2r

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Hardware Offsets Selfishness

Approach #1: use different type of guarantee

Theorem: [Roughgarden/Tardos 00] for every

network: ≤

Also: M/M/1 fns (ℓ(x)=1/(u-x), u = capacity) ⇒

≤ Nash cost at rate r

pt cost at rate 2r

Nash w/capacities 2u

pt w/capacities u

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Linear Latency Functions

Approach #2: restrict class of allowable

latency functions

Def: linear latency fn is of form ℓe(x)=aex+be Theorem: [Roughgarden/Tardos 00] for every

network with linear latency fns:

≤ 4/3 × cost of Nash flow cost of

pt flow

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Sources of Inefficiency

Corollary of previous Theorem:

For linear latency fns, worst Nash/OPT

ratio is realized in a two-link network!

simple explanation for worst inefficiency

– confronted w/two routes, selfish users

vercongest one of them

s t x 1

1

½ ½

Cost of Nash = 1
Cost of OPT = ¾

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Simple Worst-Case Networks

Theorem: [Roughgarden 02] fix any class of

latency fns, and the worst Nash/OPT ratio

ccurs in a two-node, two-link network.
under mild assumptions (convexity, richness)
inefficiency of Nash flows always has simple

explanation; simple networks are worst examples

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Simple Worst-Case Networks

Theorem: [Roughgarden 02] fix any class of

latency fns, and the worst Nash/OPT ratio

ccurs in a two-node, two-link network.
under mild assumptions (convexity, richness)
inefficiency of Nash flows always has simple

explanation; simple networks are worst examples

Proof Idea: Nash flows solve a certain minimization problem

not quite total latency, but close
electrical current is physical analog

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Computing the Price of Anarchy

Application: worst-case examples simple ⇒

worst-case ratio is easy to calculate

Example: polynomials with degree ≤ d,

nonnegative coeffs ⇒ price of anarchy Θ(d/log d)

s t xd 1

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Hardware Offsets Selfishness

Theorem: [Roughgarden/Tardos 00] for every

network: ≤

Corollary: networks with M/M/1 delay fns ⇒

≤ Nash cost at rate r

pt cost at rate 2r

Nash w/capacities 2u

pt w/capacities u

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Key Difficulty

Suppose f a Nash flow, f* an opt flow at twice the rate. Want to show that C(f) ≥ C(f). Note: cost of f can be written as C(f) = Σe fe• ℓe(fe) Similarly: C(f) = Σe f• ℓe(f) Problem: what is the relation between ℓe(fe) and ℓe(f*)?

e e e

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Key Trick

Idea: lower bound cost of f* using a

different set of latency fns c such that:

easy to lower bound cost of f* w.r.t. latency fns c
cost of f* w.r.t. fns c ≈ cost of f* w.r.t. fns ℓ

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Key Trick

Idea: lower bound cost of f* using a

different set of latency fns c such that:

easy to lower bound cost of f* w.r.t. latency fns c
cost of f* w.r.t. fns c ≈ cost of f* w.r.t. fns ℓ

The construction:

ℓe(fe) fe

graph of ℓ

ℓe(fe) fe

graph of c

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Lower Bounding OPT

Assume for simplicity: only one commodity.

all traffic in Nash flow has same latency, say L
cost of Nash flow easy to compute: C(f) = rL

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Lower Bounding OPT

Assume for simplicity: only one commodity.

all traffic in Nash flow has same latency, say L
cost of Nash flow easy to compute: C(f) = rL

Key observation: latency of path P w.r.t. latency fns c with no congestion is ℓP(f)

ℓe(fe) fe

path latency in Nash flow

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Lower Bounding OPT

Assume for simplicity: only one commodity.

all traffic in Nash flow has same latency, say L
cost of Nash flow easy to compute: C(f) = rL

Key observation: latency of path P w.r.t. latency fns c with no congestion is ℓP(f) ⇒ cost of f* w.r.t. c is at least 2rL = 2C(f)

ℓe(fe) fe

path latency in Nash flow

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Bounding the Overestimate

So far: cost of f* w.r.t. c is ≥ 2C(f). Claim: (will finish proof of Thm) [cost of f* w.r.t. c] - C(f*) ≤ C(f).

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Bounding the Overestimate

So far: cost of f* w.r.t. c is ≥ 2C(f). Claim: (will finish proof of Thm) [cost of f* w.r.t. c] - C(f*) ≤ C(f). Reason: difference in costs on edge e is

ℓe(fe) fe

typical value of ce(fe)fe - ℓe(fe)fe

* * * *

fe

*

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Bounding the Overestimate

So far: cost of f* w.r.t. c is ≥ 2C(f). Claim: (will finish proof of Thm) [cost of f* w.r.t. c] - C(f*) ≤ C(f). Reason: difference in costs on edge e is

⇒ ce(fe)fe - ℓe(fe)fe ≤ ℓe(fe)fe

ℓe(fe) fe

typical value of ce(fe)fe - ℓe(fe)fe

* * * *

fe

*

sum over edges to get Claim

* * * *

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Summary

Goal: prove that loss in network performance

due to selfish routing is not too large.

Problem: a Nash flow can cost

far more than an optimal flow.

Solutions:

compare Nash to opt flow with extra traffic
restrict class of allowable edge latency

functions, obtain bounded price of anarchy

s t xd 1

1 1-Є

Є

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Coping with Selfishness

Goal: design/manage networks so that selfish routing “not too bad” ⇒ adds algorithmic dimension Approach #1: Network design

want to avoid Braess’s Paradox

Results: [Roughgarden FOCS ‘01]

Braess’s Paradox can be arbitrarily severe in

larger networks, hard to efficiently detect

also [Lin/Roughgarden/Tardos, in prep]

s t

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Coping with Selfishness

Approach #2: Stackelberg routing

some traffic routed centrally, selfish users react

to congestion accordingly

[Roughgarden STOC ‘01]: Stackelberg routing can

dramatically improve over the Nash flow

Approach #3: Edge pricing

use economic incentives (taxes) to influence

selfish behavior

[Cole/Dodis/Roughgarden EC ‘03 + STOC ‘03]:

explore this idea for selfish routing

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Future Research

Explore other game-theoretic environments

using an approximation framework

– [Czumaj/Krysta/Voecking STOC ‘02], [Vetta FOCS ‘02], etc.

Approximation algorithms for network design

– also interesting without game-theoretic constraints

– [Kumar/Gupta/Roughgarden FOCS ‘02] – [Gupta/Kumar/Roughgarden STOC ‘03]

Algorithms for key game-theoretic concepts

– Nash/market equilibria (e.g., [Devanur et al FOCS ‘02])

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Extensions

Fact: positive results continue to hold for:

approximate Nash flows [RT00]

– users route on approximately min-latency paths

finitely many agents, splittable flow [RT00]

– weakens assumption that agents are small

“nonatomic congestion games”, games